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Coarse-Grained Energy Balance For Gas-Particle Flows

1

Kapil Agrawal, William Holloway & S. SundaresanPrinceton University

NETL 2012 Conference on Multiphase Flow Science Morgantown, WV May 22, 2012

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Simulation of Gas-Particle Flows Across Length Scales

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MICRO-SCALE~ 50μm-mm MESO-SCALE

~ mm-cm

φ

MACRO-SCALE~ cm-m

Volume-averaged hydrodynamic models for fluid and particle

phases.

Filtered volume-averaged hydrodynamic models for fluid

and particle phases.

Newton’s equations of motion for each particle, Navier-Stokes equations for the fluid flow in

the interstices.

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Development of Filtered Two-Fluid Models

• Hydrodynamics (Igci et al, 2011)

- Filtered drag *

- Filtered pressure

- Filtered viscosity

3*Li & Kwauk, 1994; Parmentier et. al, 2011

• Reacting flows (Holloway & Sundaresan, 2012)

- Filtered reaction rate

• Thermal energy & interphase transport (present work)

- Filtered energy dispersion

- Filtered interphase heat/mass transfer

- Filtered scalar dispersion (no interphase transport)

• Helium Tracer/Solid Particle Tracer

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Two-Fluid Model Equations

4

∂

∂t(ρsφsvs) +∇ · (ρsφsvsvs) = −∇ · σs − φs∇ · σg + f + ρsφsg

∂

∂t(ρgφgvg) +∇ · (ρgφgvgvg) = −φg∇ · σg − f + ρgφgg

Continuity

Momentum

Thermal EnergyρsCps

�∂

∂t(φsTs) +∇ · (φsvsTs)

�= ∇ · (ks∇Ts) + γ(Ts − Tg)

ρgCpg

�∂

∂t(φgTg) +∇ · (φgvgTg)

�= ∇ · (kg∇Tg)− γ(Ts − Tg)

Gunn (1978)

Wen & Yu (1966)

Granular kinetic theory

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Filtered Equations

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Filter φs(x, t) =

�

V∞

G(x,y)φs(x, t)dy

�αs(x, t) =

�

V∞

G(x,y)αs(x, t)φs(x, t)dy�

V∞

G(x,y)φs(x, t)dyFavre Filter

�

V∞

G(x,y)(x, t)dy = 1

ρsCps

�∂

∂t(φs

�Ts) +∇ · (φs�vs�Ts)

�= ∇ · (ks∇Ts) + γ(Ts − Tg) + ρsCps∇ ·

�φs�vs

�Ts − φs�vsTs

�

Filtered Solids Thermal Energy Equation

‘dispersive’ flux

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Applying Mean Temperature Gradient

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Temperature Gradient

Thermal Energy

Ts = T�

s + θ(x)∂θ

∂x= χ = constant

Tg = T�

g + θ(x)

ρsCps

�∂

∂t(φsT

�

s) +∇ · (φsvsT�

s)

�= ∇ · (ks∇T

�

s) + γ(T�

s − T�

g) + χ∂ks∂x

− ρsCpsφsvsxχ

ρgCpg

�∂

∂t(φgT

�

g) +∇ · (φgvgT�

g)

�= ∇ · (kg∇T

�

g)− γ(T�

s − T�

g) + χ∂kg∂x

− ρgCpgφgvgxχ

‘New’ source terms

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∂θ

∂x= χ

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Applying Heat Sources/Sinks

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Thermal Energy

‘New’ source/sink terms

ρsCps

�∂

∂t(φsTs) +∇ · (φsvsTs)

�= ∇ · (ks∇Ts) + γ(Ts − Tg) + Q̇sφs

ρgCpg

�∂

∂t(φgTg) +∇ · (φgvgTg)

�= ∇ · (kg∇Tg)− γ(Ts − Tg)− Q̇gφg

Solids Heat Source

Gas Heat Sink

Q̇gφg = Q̇sφs

φg

φg

�

DQ̇gφgdV = Q̇sφsV

Q̇sφs

�

DQ̇sφsdV = Q̇sφsV

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φs

φs

φg

φg

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Simulations and Filtering Procedure

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Snapshot of solids volume fraction field obtained from highly resolved simulations.

Shaded squares illustrate regions over which a filtering operation is performed.

2-D periodic domain with mean vertical pressure drop in gas phase set to balance weight of mixture.

Computational Domain: 32cm x 32cmDiscretization: 256 x 256 cells

Particle diameter: 75μm

Simulations are for FCC particles in air.Results are suitably scaled so that they are

applicable to other systems.

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Filtered Solids Thermal Dispersion

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Curves correspond to different filter sizes.

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

160

180

200

!s

1cm2cm3cm4cm5cm6cm7cm8cm

FCC particles in air.

αfilt =kfiltρsCps

=φs

��vsx

�Ts − �vsxTs

�

∂�Ts

∂x

��vsx

�Ts − �vsxTs

�

∂�Ts

∂x

cm2/s

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Scaled Filtered Solids Thermal Dispersion

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Dimensionless filter size shown in legend

∆

v2t /g

0 0.1 0.2 0.3 0.4 0.5 0.60

0.05

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0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

!s

4.276.418.5410.6512.8114.9417.08

δ = −4/3

(Fr∆)−1

αfilt�v3t /g

� = φs h(φs)(Fr∆)δ

h(φs)

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Filtered Gas Thermal Dispersion

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Dimensionless filter size shown in legend

∆

v2t /g

Ratio of Gas to Solids Filtered Dispersion

0.1 0.2 0.3 0.4 0.5 0.60

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1

!s

4.276.418.5410.6512.8114.9417.08

δ = −4/3

�h(φs)

�g�

h(φs)�s

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Filtered Interphase Heat Transfer Coefficient

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Curves correspond to different filter sizes.

γfiltγ(φs, �vslip)

0 0.1 0.2 0.3 0.4 0.50

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1

!s

0.25cm0.5cm1cm2cm4cm8cm

Dong et. al (2008); Kashyap & Gidaspow (2010,2011)

γfilt =γ(Ts − Tg)

(�Ts −�Tg)

1−�

γfiltγ(φs, �vslip)

�

min

∼�1 +

1�(Fr∆)

−1 − (Fr∆res)−1

�+

�−0.1

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Summary

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• Filtered dispersion coefficient

|�S| ∼ ∆2/3

• Filtered interphase mass/heat transfer coefficient

• Explore alternate scaling

• Relate momentum dispersion to energy/scalar dispersion and interphase mass/heat transfer

Next Steps

• Impact of going from 2-D to 3-D

- Results for hydrodynamics suggest the effect is small (Igci et. al, 2011)

αfilt�v3t /g

� = φs h(φs)(Fr∆)−4/3

αfilt

∆2|�S|= G(φs)

1− γfiltγ(φs, �vslip)

∼ f(φs)

�1 +

1�(Fr∆)

−1 − (Fr∆res)−1

�+

�−0.1